A quadrature Kalman filter for estimating MIRT models for sequential data
Peter van Rijn
Session 2A, 12:45 - 14:15, HAGEN 2
Although numerous approaches exist for modeling sequential data in the context of item response theory (IRT; von Davier, Xu, & Carstensen, 2011), some issues still persist. A first issue concerns the sequential nature of the data. Some approaches that have been applied in a longitudinal context are problematic because they do not take into account the order of the observations. For example, the model described by te Marvelde, Glas, Van Landeghem, and Van Damme (2006) and Andrade and Tavares (2005) assumes a straightforward multivariate normal distribution for the latent variables at different measurement occasions. However, in this approach, any permutation of the measurement occasions will show the exact same fit, because the order of the observations is not taken into account.
A second issue that can be distinguished is computational tractability. Some approaches can relatively quickly become computationally intractable, because at each time point a new latent variable is introduced (Embretson, 1991; Fischer, 1989; Andrade &Tavares, 2005; te Marvelde et al., 2006). This increases the dimensionality proportional with the number of time points, and is generally referred to as the curse of dimensionality. The problem lies in the estimation, which in IRT typically is performed by maximizing the marginal likelihood. This likelihood is obtained by integrating out the latent variables. Although current estimation methods in IRT can deal with higher dimensions, in practice, the effective dimensionality of the integral (after dimension reduction techniques) cannot be larger than six or so (Cai, 2010). This problem becomes even more pertinent when a multidimensional IRT model is to be used at each time point (Rijmen, 2010; Cho, Athay, & Preacher, 2013).
A general method for estimating multidimensional item response theory (MIRT) models for longitudinal and time series data, which addresses both these issues, is presented. The method employs an expectation-maximization (EM) algorithm in which the expectation step is formed by a discrete-time Kalman filter that makes use of adaptive Gauss-Hermite quadrature to deal with nonlinearity and non-normality (Arasaratnam, Haykin, & Elliott, 2007). The use of quadrature is highly similar to marginal maximum likelihood estimation of regular MIRT models, thereby providing a natural extension of the latter method to longitudinal and time series settings. Two applications of the method to real educational data are discussed.